2017/11/06 by Biswas, Kingshook
#53C24 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.02559
Given a Moebius homeomorphism f : ∂ X → ∂ Y between boundaries of proper, geodesically complete CAT(-1) spaces X,Y, and a family of probability measures \ μx \x ∈ X on ∂ X, we describe a continuous family of extensions \fp : X → Y \1 ≤ p ≤ ∞ of f, called the hyperbolic p-barycenter maps of f. If all the measures μx have full support then for p = ∞ the map f∞ coincides with the circumcenter map f defined previously in \citebiswas5. We use this to show that if X, Y are complete, simply connected manifolds with sectional curvatures K satisfying -b2 ≤ K ≤ -1, then the circumcenter maps of f and f-1 are √(b)-bi-Lipschitz homeomorphisms which are inverses of each other. It follows that closed negatively curved manifolds with the same marked length spectrum are bi-Lipschitz homeomorphic.